This issue requires addressing the nonlocal finite contributions
corresponding to the logarithmic UV divergences. One can calculate
these nonlocal terms directly (see, e.g., [1]}, [2]}
for the heat-kernel approach that can be used for this purpose),
or using Feynman diagrams, including in the theory of massive
fields [3]} (see also [4]}, [5]}, [6]}).
In the massless limit, the nonlocal logarithmic terms correspond
the higher-derivative divergences (REF ) and have the form
[6]}
\(\Gamma ^{(1)}_{fin,\,HD}&=&\int d^4x\sqrt{-g}\,\bigg \lbrace c_1\,R_{\mu \nu \alpha \beta }\,\mbox{ln}\,\Big ( -\,\frac{\operatorname{\square }}{\mu ^2} \Big )R^{\mu \nu \alpha \beta }\nonumber \\&&\quad \,+\,\, c_2 R_{\alpha \beta }\,\mbox{ln}\,\Big ( -\,\frac{\operatorname{\square }}{\mu ^2} \Big )R^{\alpha \beta }\,+\, c_3R \,\mbox{ln}\,\Big ( -\,\frac{\operatorname{\square }}{\mu ^2} \Big )R\bigg \rbrace .\) 
